Torsion theories induced from commutative subalgebras

TL;DR

This paper explores how torsion theories induced by commutative subalgebras can be extended to larger associative algebras, providing a new framework for understanding representations of algebras like gl(n).

Contribution

It introduces a method to lift A-torsion theories to larger algebras and establishes invariants for simple modules, generalizing classical representation theory results.

Findings

Torsion theories can be lifted from commutative subalgebras to the entire algebra.

Associated prime ideals of simple modules have uniform coheight, serving as invariants.

Provides a stratification of module categories based on prime ideal coheight.

Abstract

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any A-torsion theory defined by the coheight of prime ideals is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec A have the same coheight. Hence,thecoheight of the associated prime ideals of A is an invariant of a given simple U-module. This implies a stratification of the category of $U$-modules controlled by the coheight of associated prime ideals of A. Our approach can be viewed as a generalization of the classical paper by R.Block, it allows in particular to study representations of gl(n) beyond the classical category of weight or generalized weight modules.